An important and special case of (REF ) is the Allen–Cahn equation with \(f(u)=u-u^3\) ,
which was originally introduced in [1]} to model the motion of anti-phase boundaries in crystalline solids.
The solution represents the difference between
the concentrations of two components of the alloy
and thus should be evaluated between \(-1\)  and 1,
which is guaranteed by the MBP.
With the corresponding double-well potential \(F(u)=\frac{1}{4}(u^2-1)^2\) ,
the associated energy functional (REF ) decays in time,
which reflects the energy dissipation of the phase transition process.
Both the MBP and the energy stability are also satisfied by some variants of (REF ),
such as the nonlocal Allen–Cahn equation for phase separations within long-range interactions [2]}, [3]}
and the fractional Allen–Cahn equation used to describe some anomalous diffusion processes [4]}, [5]}.
To obtain stable numerical simulations and avoid nonphysical solutions for these models,
it is highly desirable to design numerical schemes preserving effectively these two basic physical properties, the MBP and the energy dissipation law in time discrete settings.
